I have a bit of a theoretical question about random effects models and regression. If I have a set of clustered, longitudinal data (say repeated measurements of $y$ on a number of different individuals) is there really any difference, aside for adjustments to degrees of freedom, perhaps, to fitting a regression model for each individual and averaging the parameter estimates across models to get an overall average model (i.e. all fixed effects) vs. fitting a model with individual id as a random effect in which each of the individual's observations fall (i.e a mixed model)?
Thanks.